Respuesta :
Answer:
[tex]26 - \sqrt{181}[/tex] cm
Step-by-step explanation:
The volume of the box is:
V = height * length * width
V = x*(66 - 2*x)*(90 - 2*x)
V = (66*x - 2*x^2)*(90 - 2*x)
V = 5940*x - 132*x^2 - 180*x^2 + 4*x^3
V = 4*x^3 - 312*x^2 + 5940*x
where x is the length of the sides of the squares, in cm.
The mathematical problem is :
Maximize: V = 4*x^3 - 312*x^2 + 5940*x
subject to:
x > 0
2*x < 66 <=> x < 33
In the maximum, the first derivative of V, dV/dx, is equal to zero
dV/dx = 12*x^2 - 624*x + 5940
From quadratic formula
[tex]x = \frac{-b \pm \sqrt{b^2 - 4(a)(c)}}{2(a)} [/tex]
[tex]x = \frac{624 \pm \sqrt{(-624)^2 - 4(12)(5940)}}{2(12)} [/tex]
[tex]x = \frac{624 \pm \sqrt{104256}}{24} [/tex]
[tex]x = \frac{624 \pm \sqrt{2^6*3^2*181}}{24} [/tex]
[tex]x = \frac{624 \pm 8*3*\sqrt{181}}{24} [/tex]
[tex]x_1 = \frac{624 + 24*\sqrt{181}}{24} [/tex]
[tex]x_1 = 26 + \sqrt{181}[/tex]
[tex]x_2 = \frac{624 - 24*\sqrt{181}}{24} [/tex]
[tex]x_2 = 26 - \sqrt{181}[/tex]
But [tex]x_1 > 33[/tex], then is not the correct answer.
The value of x that maximizes the volume of the box is:
[tex]\mathbf{x = 26 - \sqrt{181}}[/tex]
The dimension of the paper is given as:
[tex]\mathbf{Length = 90}[/tex]
[tex]\mathbf{Width = 66}[/tex]
When the cut-out (x) is removed, we have:
[tex]\mathbf{Length = 90 - 2x}[/tex]
[tex]\mathbf{Width = 66 - 2x}[/tex]
[tex]\mathbf{Height =x}[/tex]
So, the volume is:
[tex]\mathbf{Volume = Length \times Width \times Height}[/tex]
This gives
[tex]\mathbf{V(x)= (90 - 2x) \times (66 - 2x )\times x}[/tex]
Expand
[tex]\mathbf{V(x)= 5940x -312x^2 + 4x^3}[/tex]
Differentiate
[tex]\mathbf{V'(x)= 5940 -624x + 12x^2}[/tex]
Set to 0
[tex]\mathbf{5940 -624x + 12x^2 = 0}[/tex]
Divide through by 12
[tex]\mathbf{495 -52x + x^2 = 0}[/tex]
Rewrite as:
[tex]\mathbf{ x^2-52x + 495= 0}[/tex]
Using a calculator, the possible values of x are:
[tex]\mathbf{x = 26 \pm \sqrt{181}}[/tex]
The value [tex]\mathbf{x = 26+ \sqrt{181}}[/tex], is greater than the dimension of the box.
Hence, the value of x that maximizes the volume of the box is:
[tex]\mathbf{x = 26 - \sqrt{181}}[/tex]
Read more about volumes at:
https://brainly.com/question/1578538
